2 input models provide the driving force for a simulation model. the quality of the output is no...

2

Input models provide the driving force for a simulation model. The quality of the output is no better than the quality of inputs. We will discuss the 4 steps of input model development:

Collect data from the real system Identify a probability distribution to represent the input

process Choose parameters for the distribution Evaluate the chosen distribution and parameters for

goodness of fit.

One of the biggest tasks in solving a real problem. GIGO – garbage-in-garbage-out

Suggestions that may enhance and facilitate data collection: Plan ahead: begin by a practice or pre-observing session, watch

for unusual circumstances Analyze the data as it is being collected: check adequacy Combine homogeneous data sets, e.g. successive time periods,

during the same time period on successive days Be aware of data censoring: the quantity is not observed in its

entirety, danger of leaving out long process times Check for relationship between variables, e.g. build scatter

diagram Check for autocorrelation Collect input data, not performance data

3

4

Histograms Selecting families of distribution Parameter estimation Goodness-of-fit tests Fitting a non-stationary process

5

A frequency distribution or histogram is useful in determining the shape of a distribution

The number of class intervals depends on: The number of observations The dispersion of the data Suggested: the square root of the sample size

For continuous data: Corresponds to the probability density function of a theoretical

distribution For discrete data:

Corresponds to the probability mass function If few data points are available: combine adjacent cells to

eliminate the ragged appearance of the histogram

6

7

Vehicle Arrival Example: # of vehicles arriving at an intersection between 7 am and 7:05 am was monitored for 100 random workdays.

There are ample data, so the histogram may have a cell for each possible value in the data range

Arrivals per Period Frequency

0 121 102 193 174 105 86 77 58 59 310 311 1

Same data with different interval sizes

8

A family of distributions is selected based on:

The context of the input variable

Shape of the histogram

Frequently encountered distributions:

Easier to analyze: exponential, normal and Poisson

Harder to analyze: beta, gamma and Weibull

9

Use the physical basis of the distribution as a guide, for example: Binomial: # of successes in n trials Poisson: # of independent events that occur in a fixed amount of

time or space Normal: dist’n of a process that is the sum of a number of

component processes Exponential: time between independent events, or a process time

that is memoryless Weibull: time to failure for components Discrete or continuous uniform: models complete uncertainty Triangular: a process for which only the minimum, most likely,

and maximum values are known Empirical: resamples from the actual data collected

Page 313

10

Remember the physical characteristics of the process

Is the process naturally discrete or continuous valued?

Is it bounded?

No “true” distribution for any stochastic input process

Goal: obtain a good approximation

11

Q-Q plot is a useful tool for evaluating distribution fit If X is a random variable with cdf F, then the q-quantile of X is the

such that

When F has an inverse, = F-1(q)

Let {xi, i = 1,2, …., n} be a sample of data from X and {yj, j = 1,2, …, n} be the observations in ascending order:

where j is the ranking or order number

, for 0 1F( ) P(X ) q q

1 0.5is approximately -

j

j -y F

n

12

The plot of yj versus F-1( (j-0.5)/n) is

Approximately a straight line if F is a member of an appropriate

family of distributions

The line has slope 1 if F is a member of an appropriate family of

distributions with appropriate parameter values

13

Example: Check whether the door installation times follows a normal distribution.

The observations are now ordered from smallest to largest:

yj are plotted versus F-1( (j-0.5)/n) where F has a normal distribution with the sample mean (99.99 sec) and sample variance (0.28322 sec2)

j Value j Value j Value1 99.55 6 99.98 11 100.262 99.56 7 100.02 12 100.273 99.62 8 100.06 13 100.334 99.65 9 100.17 14 100.415 99.79 10 100.23 15 100.47

Page 317

14

Example (continued): Check whether the door installation times follow a normal distribution.

Straight line, supporting the hypothesis of a

normal distribution

Superimposed density function of

the normal distribution

15

Consider the following while evaluating the linearity of a q-q plot: The observed values never fall exactly on a straight line The ordered values are ranked and hence not independent,

unlikely for the points to be scattered about the line Variance of the extremes is higher than the middle. Linearity of

the points in the middle of the plot is more important. Q-Q plot can also be used to check homogeneity

Check whether a single distribution can represent both sample sets

Plotting the order values of the two data samples against each other

16

Parameter Estimation [Identifying the distribution]

Next step after selecting a family of distributions If observations in a sample of size n are X1, X2, …, Xn (discrete

or continuous), the sample mean and variance are:

1 1

2221

n

XnXS

n

XX

n

i i

n

i i

17

Parameter Estimation [Identifying the distribution]

Vehicle Arrival Example (continued): Table in the histogram example on slide 6 (Table 9.1 in book) can be analyzed to obtain:

The sample mean and variance are

The histogram suggests X to have a Possion distribution However, note that sample mean is not equal to sample variance. Reason: each estimator is a random variable, is not perfect.

k

j jj

k

j jj XfXf

XfXfn

1

2

1

2211

2080 and ,364 and

,...,1,10,0,12,100

63.799

)64.3(*1002080

3.64100

364

22

S

X

Table 9.1 -> Page 313

18

Goodness-of-Fit Tests [Identifying the distribution]

Conduct hypothesis testing on input data distribution using: Chi-square test

No single correct distribution in a real application exists. If very little data are available, it is unlikely to reject any candidate

distributions If a lot of data are available, it is likely to reject all candidate

distributions

Page 326

19

Chi-Square test [Goodness-of-Fit Tests]

Intuition: comparing the histogram of the data to the shape of the candidate density or mass function

Valid for large sample sizes when parameters are estimated by maximum likelihood

By arranging the n observations into a set of k class intervals or cells, the test statistics is:

which approximately follows the chi-square distribution with k-s-1 degrees of freedom, where s = # of parameters of the hypothesized distribution estimated by the sample statistics.

k

i i

ii

E

EO

1

220

)(

Observed Frequency

Expected Frequency

Ei = n*pi

where pi is the theoretical prob. of the ith interval.

Suggested Minimum = 5

20

Chi-Square test [Goodness-of-Fit Tests]

The hypothesis of a chi-square test is:

H0: The random variable, X, conforms to the distributional assumption with the parameter(s) given by the

estimate(s).

H1: The random variable X does not conform.

If the distribution tested is discrete and combining adjacent cell is not required (so that Ei > minimum requirement):

Each value of the random variable should be a class interval, unless combining is necessary, and ) x P(X ) p(x p iii

21

Chi-Square test [Goodness-of-Fit Tests]

If the distribution tested is continuous:

where ai-1 and ai are the endpoints of the ith class interval

and f(x) is the assumed pdf, F(x) is the assumed cdf. Recommended number of class intervals (k):

Caution: Different grouping of data (i.e., k) can affect the hypothesis testing result.

)()( )( 11

ii

a

ai aFaFdxxf pi

i

Sample Size, n Number of Class Intervals, k

20 Do not use the chi-square test

50 5 to 10

100 10 to 20

> 100 n1/2 to n/5

22

Chi-Square test [Goodness-of-Fit Tests]

Vehicle Arrival Example (continued): H0: the random variable is Poisson distributed.

H1: the random variable is not Poisson distributed.

Degree of freedom is k-s-1 = 7-1-1 = 5, hence, the hypothesis is rejected at the 0.05 level of significance.

!

)(

x

en

xnpEx

i

x i Observed Frequency, Oi Expected Frequency, Ei (Oi - Ei)2/Ei

0 12 2.61 10 9.62 19 17.4 0.153 17 21.1 0.84 10 19.2 4.415 8 14.0 2.576 7 8.5 0.267 5 4.48 5 2.09 3 0.8

10 3 0.3> 11 1 0.1

100 100.0 27.68

7.87

11.62 Combined because of min Ei

1.1168.27 25,05.0

20

23

p-Values and “Best Fits”[Goodness-of-Fit Tests]

p-value for the test statistics The significance level at which one would just reject H0 for the

given test statistic value. A measure of fit, the larger the better Large p-value: good fit Small p-value: poor fit

Vehicle Arrival Example (cont.): H0: data is Possion

Test statistics: , with 5 degrees of freedom p-value = 0.00004, meaning we would reject H0 with 0.00004

significance level, hence Poisson is a poor fit.

68.2720

24

p-Values and “Best Fits”[Goodness-of-Fit Tests]

Many software use p-value as the ranking measure to automatically determine the “best fit”. Things to be cautious about: Software may not know about the physical basis of the data,

distribution families it suggests may be inappropriate. Close conformance to the data does not always lead to the most

appropriate input model. p-value does not say much about where the lack of fit occurs

Recommended: always inspect the automatic selection using graphical methods.

25

Selecting Model without Data

If data is not available, some possible sources to obtain information about the process are: Engineering data: often product or process has performance

ratings provided by the manufacturer or company rules specify time or production standards.

Expert option: people who are experienced with the process or similar processes, often, they can provide optimistic, pessimistic and most-likely times, and they may know the variability as well.

Physical or conventional limitations: physical limits on performance, limits or bounds that narrow the range of the input process.

The nature of the process. The uniform, triangular, and beta distributions are often used

as input models.

26

Selecting Model without Data

Example: Production planning simulation. Input of sales volume of various products is required, salesperson

of product XYZ says that: No fewer than 1,000 units and no more than 5,000 units will be

sold. Given her experience, she believes there is a 90% chance of

selling more than 2,000 units, a 25% chance of selling more than 2,500 units, and only a 1% chance of selling more than 4,500 units.

Translating these information into a cumulative probability of being less than or equal to those goals for simulation input:

i Interval (Sales) Cumulative Frequency, ci

1 1000 x 2000 0.10

2 2000 < x 3000 0.75

3 3000 < x 4000 0.99

4 4000 < x 5000 1.00

Example 9.17: Page 336